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Diffstat (limited to 'libs/math/doc/html/math_toolkit/dist_ref')
36 files changed, 849 insertions, 206 deletions
diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dist_algorithms.html b/libs/math/doc/html/math_toolkit/dist_ref/dist_algorithms.html index 5fa251e87..16d39506b 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dist_algorithms.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dist_algorithms.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Distribution Algorithms</title> <link rel="stylesheet" href="../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dist_ref.html" title="Statistical Distributions Reference"> <link rel="prev" href="dists/weibull_dist.html" title="Weibull Distribution"> <link rel="next" href="../future.html" title="Extras/Future Directions"> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists.html b/libs/math/doc/html/math_toolkit/dist_ref/dists.html index 1cf1a59f1..d309622be 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists.html @@ -3,11 +3,11 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Distributions</title> <link rel="stylesheet" href="../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dist_ref.html" title="Statistical Distributions Reference"> <link rel="prev" href="nmp.html" title="Non-Member Properties"> -<link rel="next" href="dists/bernoulli_dist.html" title="Bernoulli Distribution"> +<link rel="next" href="dists/arcine_dist.html" title="Arcsine Distribution"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> @@ -20,13 +20,14 @@ </tr></table> <hr> <div class="spirit-nav"> -<a accesskey="p" href="nmp.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dist_ref.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="dists/bernoulli_dist.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> +<a accesskey="p" href="nmp.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dist_ref.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="dists/arcine_dist.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section"> <div class="titlepage"><div><div><h3 class="title"> <a name="math_toolkit.dist_ref.dists"></a><a class="link" href="dists.html" title="Distributions">Distributions</a> </h3></div></div></div> -<div class="toc"><dl class="toc"> +<div class="toc"><dl> +<dt><span class="section"><a href="dists/arcine_dist.html">Arcsine Distribution</a></span></dt> <dt><span class="section"><a href="dists/bernoulli_dist.html">Bernoulli Distribution</a></span></dt> <dt><span class="section"><a href="dists/beta_dist.html">Beta Distribution</a></span></dt> @@ -97,7 +98,7 @@ </tr></table> <hr> <div class="spirit-nav"> -<a accesskey="p" href="nmp.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dist_ref.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" 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http-equiv="Content-Type" content="text/html; charset=US-ASCII"> +<title>Arcsine Distribution</title> +<link rel="stylesheet" href="../../../math.css" type="text/css"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> +<link rel="up" href="../dists.html" title="Distributions"> +<link rel="prev" href="../dists.html" title="Distributions"> +<link rel="next" href="bernoulli_dist.html" title="Bernoulli Distribution"> +</head> +<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> +<table cellpadding="2" width="100%"><tr> +<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> +<td align="center"><a href="../../../../../../../index.html">Home</a></td> +<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> +<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> +<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> +<td align="center"><a href="../../../../../../../more/index.htm">More</a></td> +</tr></table> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="../dists.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="bernoulli_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +<div class="section"> +<div class="titlepage"><div><div><h4 class="title"> +<a name="math_toolkit.dist_ref.dists.arcine_dist"></a><a class="link" href="arcine_dist.html" title="Arcsine Distribution">Arcsine Distribution</a> +</h4></div></div></div> +<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">arcsine</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> +<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> + + <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> + <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> +<span class="keyword">class</span> <span class="identifier">arcsine_distribution</span><span class="special">;</span> + +<span class="keyword">typedef</span> <span class="identifier">arcsine_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">arcsine</span><span class="special">;</span> <span class="comment">// double precision standard arcsine distribution [0,1].</span> + +<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> +<span class="keyword">class</span> <span class="identifier">arcsine_distribution</span> +<span class="special">{</span> +<span class="keyword">public</span><span class="special">:</span> + <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> + <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> + + <span class="comment">// Constructor from two range parameters, x_min and x_max:</span> + <span class="identifier">arcsine_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">);</span> + + <span class="comment">// Range Parameter accessors:</span> + <span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> + <span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> +<span class="special">};</span> +<span class="special">}}</span> <span class="comment">// namespaces</span> +</pre> +<p> + The class type <code class="computeroutput"><span class="identifier">arcsine_distribution</span></code> + represents an <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">arcsine</a> + <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability + distribution function</a>. The arcsine distribution is named because + its CDF uses the inverse sin<sup>-1</sup> or arcsine. + </p> +<p> + This is implemented as a generalized version with support from <span class="emphasis"><em>x_min</em></span> + to <span class="emphasis"><em>x_max</em></span> providing the 'standard arcsine distribution' + as default with <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max = 1</em></span>. + (A few make other choices for 'standard'). + </p> +<p> + The arcsine distribution is generalized to include any bounded support + <span class="emphasis"><em>a <= x <= b</em></span> by <a href="http://reference.wolfram.com/language/ref/ArcSinDistribution.html" target="_top">Wolfram</a> + and <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">Wikipedia</a>, + but also using <span class="emphasis"><em>location</em></span> and <span class="emphasis"><em>scale</em></span> + parameters by <a href="http://www.math.uah.edu/stat/index.html" target="_top">Virtual + Laboratories in Probability and Statistics</a> <a href="http://www.math.uah.edu/stat/special/Arcsine.html" target="_top">Arcsine + distribution</a>. The end-point version is simpler and more obvious, + so we implement that. If desired, <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">this</a> + outlines how the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta + Distribution</a> can be used to add a shape factor. + </p> +<p> + The <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability + density function PDF</a> for the <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">arcsine + distribution</a> defined on the interval [<span class="emphasis"><em>x_min, x_max</em></span>] + is given by: + </p> +<p> +     f(x; x_min, x_max) = 1 /(π⋅√((x - x_min)⋅(x_max - x)) + </p> +<p> + For example, <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> + arcsine distribution, from input of + </p> +<pre class="programlisting"><span class="identifier">N</span><span class="special">[</span><span class="identifier">PDF</span><span class="special">[</span><span class="identifier">arcsinedistribution</span><span class="special">[</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">],</span> <span class="number">0.5</span><span class="special">],</span> <span class="number">50</span><span class="special">]</span> +</pre> +<p> + computes the PDF value + </p> +<pre class="programlisting"><span class="number">0.63661977236758134307553505349005744813783858296183</span> +</pre> +<p> + The Probability Density Functions (PDF) of generalized arcsine distributions + are symmetric U-shaped curves, centered on <span class="emphasis"><em>(x_max - x_min)/2</em></span>, + highest (infinite) near the two extrema, and quite flat over the central + region. + </p> +<p> + If random variate <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_min</em></span> + or <span class="emphasis"><em>x_max</em></span>, then the PDF is infinity. If random variate + <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_min</em></span> then the CDF is zero. + If random variate <span class="emphasis"><em>x</em></span> is <span class="emphasis"><em>x_max</em></span> + then the CDF is unity. + </p> +<p> + The 'Standard' (0, 1) arcsine distribution is shown in blue and some generalized + examples with other <span class="emphasis"><em>x</em></span> ranges. + </p> +<p> + <span class="inlinemediaobject"><img src="../../../../graphs/arcsine_pdf.svg" align="middle"></span> + </p> +<p> + The Cumulative Distribution Function CDF is defined as + </p> +<p> +     F(x) = 2⋅arcsin(√((x-x_min)/(x_max - x))) / π + </p> +<p> + <span class="inlinemediaobject"><img src="../../../../graphs/arcsine_cdf.svg" align="middle"></span> + </p> +<h6> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h0"></a> + <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.constructor"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.constructor">Constructor</a> + </h6> +<pre class="programlisting"><span class="identifier">arcsine_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">);</span> +</pre> +<p> + constructs an arcsine distribution with range parameters <span class="emphasis"><em>x_min</em></span> + and <span class="emphasis"><em>x_max</em></span>. + </p> +<p> + Requires <span class="emphasis"><em>x_min < x_max</em></span>, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> + is called. + </p> +<p> + For example: + </p> +<pre class="programlisting"><span class="identifier">arcsine_distribution</span><span class="special"><></span> <span class="identifier">myarcsine</span><span class="special">(-</span><span class="number">2</span><span class="special">,</span> <span class="number">4</span><span class="special">);</span> +</pre> +<p> + constructs an arcsine distribution with <span class="emphasis"><em>x_min = -2</em></span> + and <span class="emphasis"><em>x_max = 4</em></span>. + </p> +<p> + Default values of <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max = + 1</em></span> and a <code class="computeroutput"> <span class="keyword">typedef</span> <span class="identifier">arcsine_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">arcsine</span><span class="special">;</span></code> + mean that + </p> +<pre class="programlisting"><span class="identifier">arcsine</span> <span class="identifier">as</span><span class="special">;</span> +</pre> +<p> + constructs a 'Standard 01' arcsine distribution. + </p> +<h6> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h1"></a> + <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.parameter_accessors"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.parameter_accessors">Parameter + Accessors</a> + </h6> +<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">x_min</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> +<span class="identifier">RealType</span> <span class="identifier">x_max</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span> +</pre> +<p> + Return the parameter <span class="emphasis"><em>x_min</em></span> or <span class="emphasis"><em>x_max</em></span> + from which this distribution was constructed. + </p> +<p> + So, for example: + </p> +<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">arcsine_distribution</span><span class="special">;</span> + +<span class="identifier">arcsine_distribution</span><span class="special"><></span> <span class="identifier">as</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">5</span><span class="special">);</span> <span class="comment">// Cconstructs a double arcsine distribution.</span> +<span class="identifier">assert</span><span class="special">(</span><span class="identifier">as</span><span class="special">.</span><span class="identifier">x_min</span><span class="special">()</span> <span class="special">==</span> <span class="number">2.</span><span class="special">);</span> <span class="comment">// as.x_min() returns 2.</span> +<span class="identifier">assert</span><span class="special">(</span><span class="identifier">as</span><span class="special">.</span><span class="identifier">x_max</span><span class="special">()</span> <span class="special">==</span> <span class="number">5.</span><span class="special">);</span> <span class="comment">// as.x_max() returns 5.</span> +</pre> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h2"></a> + <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.non_member_accessor_functions"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.non_member_accessor_functions">Non-member + Accessor Functions</a> + </h5> +<p> + All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor + functions</a> that are generic to all distributions are supported: + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>, + <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. + </p> +<p> + The formulae for calculating these are shown in the table below, and at + <a href="http://mathworld.wolfram.com/arcsineDistribution.html" target="_top">Wolfram + Mathworld</a>. + </p> +<div class="note"><table border="0" summary="Note"> +<tr> +<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td> +<th align="left">Note</th> +</tr> +<tr><td align="left" valign="top"><p> + There are always <span class="bold"><strong>two</strong></span> values for the + <span class="bold"><strong>mode</strong></span>, at <span class="emphasis"><em>x_min</em></span> + and at <span class="emphasis"><em>x_max</em></span>, default 0 and 1, so instead we raise + the exception <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. + At these extrema, the PDFs are infinite, and the CDFs zero or unity. + </p></td></tr> +</table></div> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h3"></a> + <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.applications"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.applications">Applications</a> + </h5> +<p> + The arcsine distribution is useful to describe <a href="http://en.wikipedia.org/wiki/Random_walk" target="_top">Random + walks</a>, (including drunken walks) <a href="http://en.wikipedia.org/wiki/Brownian_motion" target="_top">Brownian + motion</a>, <a href="http://en.wikipedia.org/wiki/Wiener_process" target="_top">Weiner + processes</a>, <a href="http://en.wikipedia.org/wiki/Bernoulli_trial" target="_top">Bernoulli + trials</a>, and their appplication to solve stock market and other + <a href="http://en.wikipedia.org/wiki/Gambler%27s_ruin" target="_top">ruinous gambling + games</a>. + </p> +<p> + The random variate <span class="emphasis"><em>x</em></span> is constrained to <span class="emphasis"><em>x_min</em></span> + and <span class="emphasis"><em>x_max</em></span>, (for our 'standard' distribution, 0 and + 1), and is usually some fraction. For any other <span class="emphasis"><em>x_min</em></span> + and <span class="emphasis"><em>x_max</em></span> a fraction can be obtained from <span class="emphasis"><em>x</em></span> + using + </p> +<p> +   fraction = (x - x_min) / (x_max - x_min) + </p> +<p> + The simplest example is tossing heads and tails with a fair coin and modelling + the risk of losing, or winning. Walkers (molecules, drunks...) moving left + or right of a centre line are another common example. + </p> +<p> + The random variate <span class="emphasis"><em>x</em></span> is the fraction of time spent + on the 'winning' side. If half the time is spent on the 'winning' side + (and so the other half on the 'losing' side) then <span class="emphasis"><em>x = 1/2</em></span>. + </p> +<p> + For large numbers of tosses, this is modelled by the (standard [0,1]) arcsine + distribution, and the PDF can be calculated thus: + </p> +<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1.</span> <span class="special">/</span> <span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.637</span> +<span class="comment">// pdf has a minimum at x = 0.5</span> +</pre> +<p> + From the plot of PDF, it is clear that <span class="emphasis"><em>x</em></span> = ½ is the + <span class="bold"><strong>minimum</strong></span> of the curve, so this is the + <span class="bold"><strong>least likely</strong></span> scenario. (This is highly + counter-intuitive, considering that fair tosses must <span class="bold"><strong>eventually</strong></span> + become equal. It turns out that <span class="emphasis"><em>eventually</em></span> is not + just very long, but <span class="bold"><strong>infinite</strong></span>!). + </p> +<p> + The <span class="bold"><strong>most likely</strong></span> scenarios are towards + the extrema where <span class="emphasis"><em>x</em></span> = 0 or <span class="emphasis"><em>x</em></span> + = 1. + </p> +<p> + If fraction of time on the left is a ¼, it is only slightly more likely + because the curve is quite flat bottomed. + </p> +<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1.</span> <span class="special">/</span> <span class="number">4</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.735</span> +</pre> +<p> + If we consider fair coin-tossing games being played for 100 days (hypothetically + continuously to be 'at-limit') the person winning after day 5 will not + change in fraction 0.144 of the cases. + </p> +<p> + We can easily compute this setting <span class="emphasis"><em>x</em></span> = 5./100 = 0.05 + </p> +<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.05</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.144</span> +</pre> +<p> + Similarly, we can compute from a fraction of 0.05 /2 = 0.025 (halved because + we are considering both winners and losers) corresponding to 1 - 0.025 + or 97.5% of the gamblers, (walkers, particles...) on the <span class="bold"><strong>same + side</strong></span> of the origin + </p> +<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1</span> <span class="special">-</span> <span class="number">0.975</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.202</span> +</pre> +<p> + (use of the complement gives a bit more clarity, and avoids potential loss + of accuracy when <span class="emphasis"><em>x</em></span> is close to unity, see <a class="link" href="../../stat_tut/overview/complements.html#why_complements">why + complements?</a>). + </p> +<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="number">2</span> <span class="special">*</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.975</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.202</span> +</pre> +<p> + or we can reverse the calculation by assuming a fraction of time on one + side, say fraction 0.2, + </p> +<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">1</span> <span class="special">-</span> <span class="number">0.2</span> <span class="special">/</span> <span class="number">2</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.976</span> + +<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.2</span> <span class="special">/</span> <span class="number">2</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span> <span class="comment">// 0.976</span> +</pre> +<p> + <span class="bold"><strong>Summary</strong></span>: Every time we toss, the odds + are equal, so on average we have the same change of winning and losing. + </p> +<p> + But this is <span class="bold"><strong>not true</strong></span> for an an individual + game where one will be <span class="bold"><strong>mostly in a bad or good patch</strong></span>. + </p> +<p> + This is quite counter-intuitive to most people, but the mathematics is + clear, and gamblers continue to provide proof. + </p> +<p> + <span class="bold"><strong>Moral</strong></span>: if you in a losing patch, leave + the game. (Because the odds to recover to a good patch are poor). + </p> +<p> + <span class="bold"><strong>Corollary</strong></span>: Quit while you are ahead? + </p> +<p> + A working example is at <a href="../../../../../example/arcsine_example.cpp" target="_top">arcsine_example.cpp</a> + including sample output . + </p> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h4"></a> + <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.related_distributions"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.related_distributions">Related + distributions</a> + </h5> +<p> + The arcsine distribution with <span class="emphasis"><em>x_min = 0</em></span> and <span class="emphasis"><em>x_max + = 1</em></span> is special case of the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta + Distribution</a> with α = 1/2 and β = 1/2. + </p> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h5"></a> + <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.accuracy"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.accuracy">Accuracy</a> + </h5> +<p> + This distribution is implemented using sqrt, sine, cos and arc sine and + cos trigonometric functions which are normally accurate to a few <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">machine epsilon</a>. + But all values suffer from <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">loss + of significance or cancellation error</a> for values of <span class="emphasis"><em>x</em></span> + close to <span class="emphasis"><em>x_max</em></span>. For example, for a standard [0, 1] + arcsine distribution <span class="emphasis"><em>as</em></span>, the pdf is symmetric about + random variate <span class="emphasis"><em>x = 0.5</em></span> so that one would expect <code class="computeroutput"><span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.01</span><span class="special">)</span> <span class="special">==</span> + <span class="identifier">pdf</span><span class="special">(</span><span class="identifier">as</span><span class="special">,</span> <span class="number">0.99</span><span class="special">)</span></code>. But + as <span class="emphasis"><em>x</em></span> nears unity, there is increasing <a href="http://en.wikipedia.org/wiki/Loss_of_significance" target="_top">loss + of significance</a>. To counteract this, the complement versions of + CDF and quantile are implemented with alternative expressions using <span class="emphasis"><em>cos<sup>-1</sup></em></span> + instead of <span class="emphasis"><em>sin<sup>-1</sup></em></span>. Users should see <a class="link" href="../../stat_tut/overview/complements.html#why_complements">why + complements?</a> for guidance on when to avoid loss of accuracy by using + complements. + </p> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h6"></a> + <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.testing"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.testing">Testing</a> + </h5> +<p> + The results were tested against a few accurate spot values computed by + <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a>, for example: + </p> +<pre class="programlisting"><span class="identifier">N</span><span class="special">[</span><span class="identifier">PDF</span><span class="special">[</span><span class="identifier">arcsinedistribution</span><span class="special">[</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">],</span> <span class="number">0.5</span><span class="special">],</span> <span class="number">50</span><span class="special">]</span> + <span class="number">0.63661977236758134307553505349005744813783858296183</span> +</pre> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h7"></a> + <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.implementation"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.implementation">Implementation</a> + </h5> +<p> + In the following table <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> + are the parameters <span class="emphasis"><em>x_min</em></span>   and <span class="emphasis"><em>x_max</em></span>, + <span class="emphasis"><em>x</em></span> is the random variable, <span class="emphasis"><em>p</em></span> is + the probability and its complement <span class="emphasis"><em>q = 1-p</em></span>. + </p> +<div class="informaltable"><table class="table"> +<colgroup> +<col> +<col> +</colgroup> +<thead><tr> +<th> + <p> + Function + </p> + </th> +<th> + <p> + Implementation Notes + </p> + </th> +</tr></thead> +<tbody> +<tr> +<td> + <p> + support + </p> + </td> +<td> + <p> + x ∈ [a, b], default x ∈ [0, 1] + </p> + </td> +</tr> +<tr> +<td> + <p> + pdf + </p> + </td> +<td> + <p> + f(x; a, b) = 1/(π⋅√(x - a)⋅(b - x)) + </p> + </td> +</tr> +<tr> +<td> + <p> + cdf + </p> + </td> +<td> + <p> + F(x) = 2/π⋅sin<sup>-1</sup>(√(x - a) / (b - a) ) + </p> + </td> +</tr> +<tr> +<td> + <p> + cdf of complement + </p> + </td> +<td> + <p> + 2/(π⋅cos<sup>-1</sup>(√(x - a) / (b - a))) + </p> + </td> +</tr> +<tr> +<td> + <p> + quantile + </p> + </td> +<td> + <p> + -a⋅sin<sup>2</sup>(½π⋅p) + a + b⋅sin<sup>2</sup>(½π⋅p) + </p> + </td> +</tr> +<tr> +<td> + <p> + quantile from the complement + </p> + </td> +<td> + <p> + -a⋅cos<sup>2</sup>(½π⋅p) + a + b⋅cos<sup>2</sup>(½π⋅q) + </p> + </td> +</tr> +<tr> +<td> + <p> + mean + </p> + </td> +<td> + <p> + ½(a+b) + </p> + </td> +</tr> +<tr> +<td> + <p> + median + </p> + </td> +<td> + <p> + ½(a+b) + </p> + </td> +</tr> +<tr> +<td> + <p> + mode + </p> + </td> +<td> + <p> + x ∈ [a, b], so raises domain_error (returning NaN). + </p> + </td> +</tr> +<tr> +<td> + <p> + variance + </p> + </td> +<td> + <p> + (b - a)<sup>2</sup> / 8 + </p> + </td> +</tr> +<tr> +<td> + <p> + skewness + </p> + </td> +<td> + <p> + 0 + </p> + </td> +</tr> +<tr> +<td> + <p> + kurtosis excess + </p> + </td> +<td> + <p> + -3/2 + </p> + </td> +</tr> +<tr> +<td> + <p> + kurtosis + </p> + </td> +<td> + <p> + kurtosis_excess + 3 + </p> + </td> +</tr> +</tbody> +</table></div> +<p> + The quantile was calculated using an expression obtained by using <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> to invert the + formula for the CDF thus + </p> +<pre class="programlisting"><span class="identifier">solve</span> <span class="special">[</span><span class="identifier">p</span> <span class="special">-</span> <span class="number">2</span><span class="special">/</span><span class="identifier">pi</span> <span class="identifier">sin</span><span class="special">^-</span><span class="number">1</span><span class="special">(</span><span class="identifier">sqrt</span><span class="special">((</span><span class="identifier">x</span><span class="special">-</span><span class="identifier">a</span><span class="special">)/(</span><span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)))</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">x</span><span class="special">]</span> +</pre> +<p> + which was interpreted as + </p> +<pre class="programlisting"><span class="identifier">Solve</span><span class="special">[</span><span class="identifier">p</span> <span class="special">-</span> <span class="special">(</span><span class="number">2</span> <span class="identifier">ArcSin</span><span class="special">[</span><span class="identifier">Sqrt</span><span class="special">[(-</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">x</span><span class="special">)/(-</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)]])/</span><span class="identifier">Pi</span> <span class="special">==</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">MaxExtraConditions</span> <span class="special">-></span> <span class="identifier">Automatic</span><span class="special">]</span> +</pre> +<p> + and produced the resulting expression + </p> +<pre class="programlisting"><span class="identifier">x</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">a</span> <span class="identifier">sin</span><span class="special">^</span><span class="number">2</span><span class="special">((</span><span class="identifier">pi</span> <span class="identifier">p</span><span class="special">)/</span><span class="number">2</span><span class="special">)+</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span> <span class="identifier">sin</span><span class="special">^</span><span class="number">2</span><span class="special">((</span><span class="identifier">pi</span> <span class="identifier">p</span><span class="special">)/</span><span class="number">2</span><span class="special">)</span> +</pre> +<p> + Thanks to Wolfram for providing this facility. + </p> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h8"></a> + <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.references"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.references">References</a> + </h5> +<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> +<li class="listitem"> + <a href="http://en.wikipedia.org/wiki/arcsine_distribution" target="_top">Wikipedia + arcsine distribution</a> + </li> +<li class="listitem"> + <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia + Beta distribution</a> + </li> +<li class="listitem"> + <a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram + MathWorld</a> + </li> +<li class="listitem"> + <a href="http://www.wolframalpha.com/" target="_top">Wolfram Alpha</a> + </li> +</ul></div> +<h5> +<a name="math_toolkit.dist_ref.dists.arcine_dist.h9"></a> + <span class="phrase"><a name="math_toolkit.dist_ref.dists.arcine_dist.sources"></a></span><a class="link" href="arcine_dist.html#math_toolkit.dist_ref.dists.arcine_dist.sources">Sources</a> + </h5> +<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> +<li class="listitem"> + <a href="http://estebanmoro.org/2009/04/the-probability-of-going-through-a-bad-patch" target="_top">The + probability of going through a bad patch</a> Esteban Moro's Blog. + </li> +<li class="listitem"> + <a href="http://www.gotohaggstrom.com/What%20do%20schmucks%20and%20the%20arc%20sine%20law%20have%20in%20common.pdf" target="_top">What + soschumcks and the arc sine have in common</a> Peter Haggstrom. + </li> +<li class="listitem"> + <a href="http://www.math.uah.edu/stat/special/Arcsine.html" target="_top">arcsine + distribution</a>. + </li> +<li class="listitem"> + <a href="http://reference.wolfram.com/language/ref/ArcSinDistribution.html" target="_top">Wolfram + reference arcsine examples</a>. + </li> +<li class="listitem"> + <a href="http://www.math.harvard.edu/library/sternberg/slides/1180908.pdf" target="_top">Shlomo + Sternberg slides</a>. + </li> +</ul></div> +</div> +<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> +<td align="left"></td> +<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, + Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert + Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, + Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> + Distributed under the Boost Software License, Version 1.0. (See accompanying + file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) + </p> +</div></td> +</tr></table> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="../dists.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="bernoulli_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +</body> +</html> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/bernoulli_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/bernoulli_dist.html index d8bc0819e..76cdafb06 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/bernoulli_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/bernoulli_dist.html @@ -3,10 +3,10 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Bernoulli Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> -<link rel="prev" href="../dists.html" title="Distributions"> +<link rel="prev" href="arcine_dist.html" title="Arcsine Distribution"> <link rel="next" href="beta_dist.html" title="Beta Distribution"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> @@ -20,7 +20,7 @@ </tr></table> <hr> <div class="spirit-nav"> -<a accesskey="p" href="../dists.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="beta_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> +<a accesskey="p" href="arcine_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="beta_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section"> <div class="titlepage"><div><div><h4 class="title"> @@ -72,14 +72,14 @@ density function pdf</a> varies with the outcome of the single trial: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/bernoulli_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/bernoulli_pdf.svg" align="middle"></span> </p> <p> and the <a href="http://en.wikipedia.org/wiki/Cumulative_Distribution_Function" target="_top">Cumulative distribution function</a> </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/bernoulli_cdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/bernoulli_cdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.bernoulli_dist.h0"></a> @@ -350,7 +350,7 @@ </tr></table> <hr> <div class="spirit-nav"> -<a accesskey="p" href="../dists.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="beta_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> +<a accesskey="p" href="arcine_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="beta_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html index 7ca1ca0d5..3459c6265 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/beta_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Beta Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="bernoulli_dist.html" title="Bernoulli Distribution"> <link rel="next" href="binomial_dist.html" title="Binomial Distribution"> @@ -116,7 +116,7 @@ might be approximated by a symmetrical triangular distribution. </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/beta_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/beta_pdf.svg" align="middle"></span> </p> <p> If α = β = 1, then it is a __space <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform @@ -466,7 +466,7 @@ </td> <td> <p> - <span class="inlinemediaobject"><img src="../../../../equations/beta_dist_kurtosis.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/beta_dist_kurtosis.svg"></span> </p> </td> </tr> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/binomial_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/binomial_dist.html index f2e7020ea..57ac34667 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/binomial_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/binomial_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Binomial Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="beta_dist.html" title="Beta Distribution"> <link rel="next" href="cauchy_dist.html" title="Cauchy-Lorentz Distribution"> @@ -105,7 +105,7 @@ The PDF for the binomial distribution is given by: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/binomial_ref2.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/binomial_ref2.svg"></span> </p> <p> The following two graphs illustrate how the PDF changes depending upon @@ -113,14 +113,14 @@ fixed at 0.5, and vary the sample size: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/binomial_pdf_1.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/binomial_pdf_1.svg" align="middle"></span> </p> <p> Alternatively, we can keep the sample size fixed at N=20 and vary the success fraction <span class="emphasis"><em>p</em></span>: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/binomial_pdf_2.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/binomial_pdf_2.svg" align="middle"></span> </p> <div class="caution"><table border="0" summary="Caution"> <tr> @@ -673,7 +673,7 @@ if <sub>n</sub>C<sub>k </sub> is the binomial coefficient of a and b, then we have: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/binomial_ref1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/binomial_ref1.svg"></span> </p> <p> Which can be evaluated as <code class="computeroutput"><span class="identifier">ibeta_derivative</span><span class="special">(</span><span class="identifier">k</span><span class="special">+</span><span class="number">1</span><span class="special">,</span> <span class="identifier">n</span><span class="special">-</span><span class="identifier">k</span><span class="special">+</span><span class="number">1</span><span class="special">,</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">/</span> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/cauchy_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/cauchy_dist.html index f17f3bc43..3088e031a 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/cauchy_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/cauchy_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Cauchy-Lorentz Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="binomial_dist.html" title="Binomial Distribution"> <link rel="next" href="chi_squared_dist.html" title="Chi Squared Distribution"> @@ -55,7 +55,7 @@ distribution function PDF</a> given by: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/cauchy_ref1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/cauchy_ref1.svg"></span> </p> <p> The location parameter x<sub>0</sub>   is the location of the peak of the distribution @@ -73,14 +73,14 @@ changes: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/cauchy_pdf1.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/cauchy_pdf1.svg" align="middle"></span> </p> <p> While the following graph shows how the shape (scale) parameter alters the distribution: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/cauchy_pdf2.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/cauchy_pdf2.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.cauchy_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/chi_squared_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/chi_squared_dist.html index e5a22af4d..97dd33791 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/chi_squared_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/chi_squared_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Chi Squared Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="cauchy_dist.html" title="Cauchy-Lorentz Distribution"> <link rel="next" href="exp_dist.html" title="Exponential Distribution"> @@ -67,7 +67,7 @@ independent, normally distributed random variables with means μ<sub>i</sub>   and variances σ<sub>i</sub><sup>2</sup>, then the random variable: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/chi_squ_ref1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/chi_squ_ref1.svg"></span> </p> <p> is distributed according to the Chi-Squared distribution. @@ -79,7 +79,7 @@ independent, normally distributed random values of ν: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/chi_squared_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/chi_squared_pdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.chi_squared_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/exp_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/exp_dist.html index 885c507a9..b1abb014d 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/exp_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/exp_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Exponential Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="chi_squared_dist.html" title="Chi Squared Distribution"> <link rel="next" href="extreme_dist.html" title="Extreme Value Distribution"> @@ -51,7 +51,7 @@ probability distribution</a> with PDF: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/exponential_dist_ref1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/exponential_dist_ref1.svg"></span> </p> <p> It is often used to model the time between independent events that happen @@ -62,7 +62,7 @@ of the rate parameter lambda: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/exponential_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/exponential_pdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.exp_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/extreme_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/extreme_dist.html index 10d8d7754..199c75f15 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/extreme_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/extreme_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Extreme Value Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="exp_dist.html" title="Exponential Distribution"> <link rel="next" href="f_dist.html" title="F Distribution"> @@ -85,13 +85,13 @@ The following graph illustrates how the PDF varies with the location parameter: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/extreme_value_pdf1.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/extreme_value_pdf1.svg" align="middle"></span> </p> <p> And this graph illustrates how the PDF varies with the shape parameter: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/extreme_value_pdf2.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/extreme_value_pdf2.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.extreme_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/f_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/f_dist.html index b9f2a989a..7cfe73954 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/f_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/f_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>F Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="extreme_dist.html" title="Extreme Value Distribution"> <link rel="next" href="gamma_dist.html" title="Gamma (and Erlang) Distribution"> @@ -65,14 +65,14 @@ PDF: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span> </p> <p> The following graph illustrates how the PDF varies depending on the two degrees of freedom parameters. </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/fisher_f_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/fisher_f_pdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.f_dist.h0"></a> @@ -175,7 +175,7 @@ The usual form of the PDF is given by: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span> </p> <p> However, that form is hard to evaluate directly without incurring diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/gamma_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/gamma_dist.html index 4b5e90583..77d91b4e2 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/gamma_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/gamma_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Gamma (and Erlang) Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="f_dist.html" title="F Distribution"> <link rel="next" href="geometric_dist.html" title="Geometric Distribution"> @@ -79,14 +79,14 @@ by the probability density function: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/gamma_dist_ref1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/gamma_dist_ref1.svg"></span> </p> <p> Sometimes an alternative formulation is used: given parameters α  = k and β  = 1 / θ, then the distribution can be defined by the PDF: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/gamma_dist_ref2.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/gamma_dist_ref2.svg"></span> </p> <p> In this form the inverse scale parameter is called a <span class="emphasis"><em>rate parameter</em></span>. @@ -102,10 +102,10 @@ varies as the parameters vary: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/gamma1_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/gamma1_pdf.svg" align="middle"></span> </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/gamma2_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/gamma2_pdf.svg" align="middle"></span> </p> <p> The <span class="bold"><strong>Erlang Distribution</strong></span> is the same as diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/geometric_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/geometric_dist.html index 25d49a877..f782bfe80 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/geometric_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/geometric_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Geometric Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="gamma_dist.html" title="Gamma (and Erlang) Distribution"> <link rel="next" href="hyperexponential_dist.html" title="Hyperexponential Distribution"> @@ -125,19 +125,19 @@ distribution as a continuous function), </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/geometric_pdf_2.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/geometric_pdf_2.svg" align="middle"></span> </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/geometric_cdf_2.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/geometric_cdf_2.svg" align="middle"></span> </p> <p> and as discrete. </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/geometric_pdf_discrete.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/geometric_pdf_discrete.svg" align="middle"></span> </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/geometric_cdf_discrete.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/geometric_cdf_discrete.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.geometric_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/hyperexponential_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/hyperexponential_dist.html index b17a9ae46..86f53b8ad 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/hyperexponential_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/hyperexponential_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Hyperexponential Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="geometric_dist.html" title="Geometric Distribution"> <link rel="next" href="hypergeometric_dist.html" title="Hypergeometric Distribution"> @@ -111,7 +111,7 @@ hyperexponential distribution is given by: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_pdf.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_pdf.svg"></span> </p> <p> The following graph illustrates the PDF of the hyperexponential distribution @@ -140,7 +140,7 @@ </li> </ol></div> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf.svg" align="middle"></span> </p> <p> Also, the following graph illustrates the PDF of the hyperexponential distribution @@ -180,7 +180,7 @@ nears the exponential distribution with parameter <span class="emphasis"><em>λ=1.5</em></span>. </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samerate.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samerate.svg" align="middle"></span> </p> <p> Finally, the following graph compares the PDF of the hyperexponential distribution @@ -202,7 +202,7 @@ </li> </ol></div> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samemean.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/hyperexponential_pdf_samemean.svg" align="middle"></span> </p> <p> As can be noted, even if the three distributions have the same mean value, @@ -232,7 +232,7 @@ often been observed to possess such a distribution (Rosin,1965). Also, the arrival of different types of customer to a single queueing station is often modeled as a hyperexponential distribution (Papadopolous et al.,1993). - Similarly, if a product manufactured in several parallel assemply lines + Similarly, if a product manufactured in several parallel assembly lines and the outputs are merged, the failure density of the overall product is likely to be hyperexponential (Trivedi,2002). </p> @@ -866,7 +866,7 @@ </tr> <tr><td align="left" valign="top"><p> As an implementation detail, this constructor uses Boost's <a href="http://www.boost.org/doc/libs/release/libs/core/doc/html/core/enable_if.html" target="_top">enable_if/disable_if - mechanism</a> to disambiguage between this and other 2-argument constructors. + mechanism</a> to disambiguate between this and other 2-argument constructors. Refer to the source code for more details. </p></td></tr> </table></div> @@ -914,7 +914,7 @@ <h5> <a name="math_toolkit.dist_ref.dists.hyperexponential_dist.h15"></a> <span class="phrase"><a name="math_toolkit.dist_ref.dists.hyperexponential_dist.construction_with_rates_iterator"></a></span><a class="link" href="hyperexponential_dist.html#math_toolkit.dist_ref.dists.hyperexponential_dist.construction_with_rates_iterator">Construction - with rates-iterators (and all phase probablities equal)</a> + with rates-iterators (and all phase probabilities equal)</a> </h5> <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">RateIterT</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">RateIterT2</span><span class="special">></span> <span class="identifier">hyperexponential_distribution</span><span class="special">(</span><span class="identifier">RateIterT</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">rate_first</span><span class="special">,</span> @@ -935,7 +935,7 @@ </tr> <tr><td align="left" valign="top"><p> As an implementation detail, this constructor uses Boost's <a href="http://www.boost.org/doc/libs/release/libs/core/doc/html/core/enable_if.html" target="_top">enable_if/disable_if - mechanism</a> to disambiguage between this and other 2-argument constructors. + mechanism</a> to disambiguate between this and other 2-argument constructors. Refer to the source code for more details. </p></td></tr> </table></div> @@ -1264,7 +1264,7 @@ </td> <td> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_pdf.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_pdf.svg"></span> </p> </td> </tr> @@ -1276,7 +1276,7 @@ </td> <td> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_cdf.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_cdf.svg"></span> </p> </td> </tr> @@ -1288,7 +1288,7 @@ </td> <td> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_ccdf.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_ccdf.svg"></span> </p> </td> </tr> @@ -1324,7 +1324,7 @@ </td> <td> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_mean.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_mean.svg"></span> </p> </td> </tr> @@ -1336,7 +1336,7 @@ </td> <td> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_variance.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_variance.svg"></span> </p> </td> </tr> @@ -1360,7 +1360,7 @@ </td> <td> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_skewness.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_skewness.svg"></span> </p> </td> </tr> @@ -1372,7 +1372,7 @@ </td> <td> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_kurtosis.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hyperexponential_kurtosis.svg"></span> </p> </td> </tr> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/hypergeometric_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/hypergeometric_dist.html index 99c2470d1..9adfc46e5 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/hypergeometric_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/hypergeometric_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Hypergeometric Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="hyperexponential_dist.html" title="Hyperexponential Distribution"> <link rel="next" href="inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution"> @@ -86,7 +86,7 @@ sample sizes constant: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/hypergeometric_pdf_1.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/hypergeometric_pdf_1.svg" align="middle"></span> </p> <p> Note that since the distribution is symmetrical in parameters <span class="emphasis"><em>n</em></span> @@ -95,7 +95,7 @@ the same graphs: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/hypergeometric_pdf_2.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/hypergeometric_pdf_2.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.hypergeometric_dist.h0"></a> @@ -105,7 +105,7 @@ <pre class="programlisting"><span class="identifier">hypergeometric_distribution</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">);</span> </pre> <p> - Constructs a hypergeometric distribution with with a population of <span class="emphasis"><em>N</em></span> + Constructs a hypergeometric distribution with a population of <span class="emphasis"><em>N</em></span> objects, of which <span class="emphasis"><em>r</em></span> are defective, and from which <span class="emphasis"><em>n</em></span> are sampled. </p> @@ -202,7 +202,7 @@ N = 110000. </p> <p> - In general for very large N, the user should expect to loose log<sub>10</sub>N decimal + In general for very large N, the user should expect to lose log<sub>10</sub>N decimal digits of precision during the calculation, with the results becoming meaningless for N >= 10<sup>15</sup>. </p> @@ -226,7 +226,7 @@ The PDF can be calculated directly using the formula: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric1.svg"></span> </p> <p> However, this can only be used directly when the largest of the factorials @@ -244,21 +244,21 @@ method for factorising a factorial into the product of prime numbers: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric2.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric2.svg"></span> </p> <p> Where p<sub>i</sub> is the i'th prime number, and e<sub>i</sub> is a small positive integer or zero, which can be calculated via: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric3.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric3.svg"></span> </p> <p> Further we can combine the factorials in the expression for the PDF to yield the PDF directly as the product of prime numbers: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric4.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric4.svg"></span> </p> <p> With this time the exponents e<sub>i</sub> being either positive, negative or zero. @@ -289,7 +289,7 @@ lgamma, or by directly combining lanczos approximations to avoid calculating via logarithms. We use the latter method, as it is usually 1 or 2 decimal digits more accurate than computing via logarithms with lgamma. However, - in this area where N > 104729, the user should expect to loose around + in this area where N > 104729, the user should expect to lose around log<sub>10</sub>N decimal digits during the calculation in the worst case. </p> <p> @@ -300,7 +300,7 @@ successive PDF values via the recurrence relations: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric5.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric5.svg"></span> </p> <p> Until we either reach the end of the distributions domain, or the next @@ -317,7 +317,7 @@ are calculated via: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric6.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric6.svg"></span> </p> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_chi_squared_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_chi_squared_dist.html index 20a9a604e..c636e97c8 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_chi_squared_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_chi_squared_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Inverse Chi Squared Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="hypergeometric_dist.html" title="Hypergeometric Distribution"> <link rel="next" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution"> @@ -177,10 +177,10 @@ distribution varies for a few values of parameters ν and ξ: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_pdf.svg" align="middle"></span> </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_cdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/inverse_chi_squared_cdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.inverse_chi_squared_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_gamma_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_gamma_dist.html index 8fd13512a..e28a3e7b8 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_gamma_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_gamma_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Inverse Gamma Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution"> <link rel="next" href="inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution"> @@ -108,10 +108,10 @@ distribution varies as the parameters vary: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gamma_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gamma_pdf.svg" align="middle"></span> </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gamma_cdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gamma_cdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.inverse_gamma_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_gaussian_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_gaussian_dist.html index 2db348156..053911f0f 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_gaussian_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/inverse_gaussian_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Inverse Gaussian (or Inverse Normal) Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution"> <link rel="next" href="laplace_dist.html" title="Laplace Distribution"> @@ -123,10 +123,10 @@ distribution varies for a few values of parameters μ and λ: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_pdf.svg" align="middle"></span> </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_cdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_cdf.svg" align="middle"></span> </p> <p> Tweedie also provided 3 other parameterisations where (μ and λ) are replaced diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/laplace_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/laplace_dist.html index e72302de0..571e61eaa 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/laplace_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/laplace_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Laplace Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution"> <link rel="next" href="logistic_dist.html" title="Logistic Distribution"> @@ -60,7 +60,7 @@ density function: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/laplace_pdf.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/laplace_pdf.svg"></span> </p> <p> The location and scale parameters are equivalent to the mean and standard @@ -72,7 +72,7 @@ of the value of the location parameter: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/laplace_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/laplace_pdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.laplace_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/logistic_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/logistic_dist.html index cecda20f7..d2a9242ac 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/logistic_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/logistic_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Logistic Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="laplace_dist.html" title="Laplace Distribution"> <link rel="next" href="lognormal_dist.html" title="Log Normal Distribution"> @@ -64,7 +64,7 @@ change: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/logistic_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/logistic_pdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.logistic_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/lognormal_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/lognormal_dist.html index abb83592c..0e2be8ff6 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/lognormal_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/lognormal_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Log Normal Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="logistic_dist.html" title="Logistic Distribution"> <link rel="next" href="negative_binomial_dist.html" title="Negative Binomial Distribution"> @@ -62,7 +62,7 @@ it is defined by the probability density function: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/lognormal_ref.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/lognormal_ref.svg"></span> </p> <p> The location and scale parameters are equivalent to the mean and standard @@ -74,13 +74,13 @@ of the value of the location parameter: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/lognormal_pdf1.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/lognormal_pdf1.svg" align="middle"></span> </p> <p> The next graph illustrates the effect of the scale parameter on the PDF: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/lognormal_pdf2.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/lognormal_pdf2.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.lognormal_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_beta_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_beta_dist.html index 032d837f4..530942e9f 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_beta_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_beta_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Noncentral Beta Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="negative_binomial_dist.html" title="Negative Binomial Distribution"> <link rel="next" href="nc_chi_squared_dist.html" title="Noncentral Chi-Squared Distribution"> @@ -72,7 +72,7 @@ is PDFs: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span> </p> <p> where P(i;λ/2) is the discrete Poisson probablity at <span class="emphasis"><em>i</em></span>, @@ -80,14 +80,14 @@ is This leads to the usual form of the CDF as: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref2.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref2.svg"></span> </p> <p> The following graph illustrates how the distribution changes for different values of λ: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/nc_beta_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/nc_beta_pdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h0"></a> @@ -321,13 +321,13 @@ is pp. 146-156. </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref3.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref3.svg"></span> </p> <p> Then either the CDF or its complement is computed using the relations: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref4.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref4.svg"></span> </p> <p> The summation is performed by starting at i = λ/2, and then recursing in @@ -384,7 +384,7 @@ is and the relation: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_beta_ref1.svg"></span> </p> <p> Quantiles are computed using a specially modified version of <a class="link" href="../../internals1/roots2.html" title="Root Finding Without Derivatives: Bisection, Bracket and TOMS748">bracket_and_solve_root</a>, diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_chi_squared_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_chi_squared_dist.html index c4d038859..3b727b392 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_chi_squared_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_chi_squared_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Noncentral Chi-Squared Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="nc_beta_dist.html" title="Noncentral Beta Distribution"> <link rel="next" href="nc_f_dist.html" title="Noncentral F Distribution"> @@ -70,7 +70,7 @@ μ<sub>i</sub> and variances σ<sub>i</sub><sup>2</sup>, then the random variable </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref1.svg"></span> </p> <p> is distributed according to the noncentral chi-squared distribution. @@ -81,7 +81,7 @@ related to the mean of the random variables X<sub>i</sub> by: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref2.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref2.svg"></span> </p> <p> (Note that some references define λ as one half of the above sum). @@ -90,7 +90,7 @@ This leads to a PDF of: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref3.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref3.svg"></span> </p> <p> where <span class="emphasis"><em>f(x;k)</em></span> is the central chi-squared distribution @@ -102,7 +102,7 @@ values of λ: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/nccs_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/nccs_pdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.nc_chi_squared_dist.h0"></a> @@ -390,7 +390,7 @@ 41, No. 2. (1992), pp. 478-482). This uses the following series representation: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref4.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref4.svg"></span> </p> <p> which requires just one call to <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a> @@ -415,7 +415,7 @@ This method uses the well known sum: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref5.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref5.svg"></span> </p> <p> Where P<sub>a</sub>(x) is the incomplete gamma function. @@ -436,7 +436,7 @@ method, given that: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref6.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref6.svg"></span> </p> <p> we can again start at the λ'th term and proceed in both directions from @@ -449,7 +449,7 @@ The PDF is computed directly using the relation: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref3.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref3.svg"></span> </p> <p> Where <span class="emphasis"><em>f(x; v)</em></span> is the PDF of the central <a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi @@ -474,7 +474,7 @@ The remaining non-member functions use the following formulas: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref7.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_chi_squ_ref7.svg"></span> </p> <p> Some analytic properties of noncentral distributions (particularly unimodality, diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_f_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_f_dist.html index 08f5610bf..aec21e27c 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_f_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_f_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Noncentral F Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="nc_chi_squared_dist.html" title="Noncentral Chi-Squared Distribution"> <link rel="next" href="nc_t_dist.html" title="Noncentral T Distribution"> @@ -72,20 +72,20 @@ random variable with <span class="emphasis"><em>v1</em></span> degrees This gives the following PDF: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_f_ref1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_f_ref1.svg"></span> </p> <p> where L<sub>a</sub><sup>b</sup>(c) is a generalised Laguerre polynomial and B(a,b) is the <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a> function, or </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_f_ref2.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_f_ref2.svg"></span> </p> <p> The following graph illustrates how the distribution changes for different values of λ: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/nc_f_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/nc_f_pdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.nc_f_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_t_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_t_dist.html index c6f521ba3..c9d9b43df 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_t_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_t_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Noncentral T Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="nc_f_dist.html" title="Noncentral F Distribution"> <link rel="next" href="normal_dist.html" title="Normal (Gaussian) Distribution"> @@ -67,7 +67,7 @@ This gives the following PDF: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref1.svg"></span> </p> <p> where <sub>1</sub>F<sub>1</sub>(a;b;x) is a confluent hypergeometric function. @@ -77,8 +77,8 @@ values of ν and δ: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/nc_t_pdf.png" align="middle"></span> - <span class="inlinemediaobject"><img src="../../../../graphs/nc_t_cdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/nc_t_pdf.svg" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/nc_t_cdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.nc_t_dist.h0"></a> @@ -300,7 +300,7 @@ when the normal distribution This uses the following formula for the CDF: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref2.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref2.svg"></span> </p> <p> Where I<sub>x</sub>(a,b) is the incomplete beta function, and Φ(x) is the normal CDF @@ -316,13 +316,13 @@ when the normal distribution therefore I<sub>x</sub>(a,b) = 1 and: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref3.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref3.svg"></span> </p> <p> From this we can easily show that: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref4.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref4.svg"></span> </p> <p> and therefore we have a means to compute either the probability or its @@ -335,7 +335,7 @@ when the normal distribution The PDF can be computed by a very similar method using: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref5.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/nc_t_ref5.svg"></span> </p> <p> Where I<sub>x</sub><sup>'</sup>(a,b) is the derivative of the incomplete beta function. diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/negative_binomial_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/negative_binomial_dist.html index 5a64924eb..3926a6cac 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/negative_binomial_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/negative_binomial_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Negative Binomial Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="lognormal_dist.html" title="Log Normal Distribution"> <link rel="next" href="nc_beta_dist.html" title="Noncentral Beta Distribution"> @@ -101,21 +101,21 @@ It has the PDF: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/neg_binomial_ref.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/neg_binomial_ref.svg"></span> </p> <p> The following graph illustrate how the PDF varies as the success fraction <span class="emphasis"><em>p</em></span> changes: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/negative_binomial_pdf_1.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/negative_binomial_pdf_1.svg" align="middle"></span> </p> <p> Alternatively, this graph shows how the shape of the PDF varies as the number of successes changes: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/negative_binomial_pdf_2.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/negative_binomial_pdf_2.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.negative_binomial_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/normal_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/normal_dist.html index 877702c6a..38ee5a6d9 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/normal_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/normal_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Normal (Gaussian) Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="nc_t_dist.html" title="Noncentral T Distribution"> <link rel="next" href="pareto.html" title="Pareto Distribution"> @@ -64,26 +64,26 @@ Given mean μ  and standard deviation σ  it has the PDF: </p> <p> -   <span class="inlinemediaobject"><img src="../../../../equations/normal_ref1.png"></span> +   <span class="inlinemediaobject"><img src="../../../../equations/normal_ref1.svg"></span> </p> <p> The variation the PDF with its parameters is illustrated in the following graph: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/normal_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/normal_pdf.svg" align="middle"></span> </p> <p> The cumulative distribution function is given by </p> <p> -   <span class="inlinemediaobject"><img src="../../../../equations/normal_cdf.png"></span> +   <span class="inlinemediaobject"><img src="../../../../equations/normal_cdf.svg"></span> </p> <p> and illustrated by this graph </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/normal_cdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/normal_cdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.normal_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/pareto.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/pareto.html index 4b993f461..266599927 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/pareto.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/pareto.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Pareto Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="normal_dist.html" title="Normal (Gaussian) Distribution"> <link rel="next" href="poisson_dist.html" title="Poisson Distribution"> @@ -71,14 +71,14 @@ β: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf1.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf1.svg" align="middle"></span> </p> <p> And this graph illustrates how the PDF varies with the shape parameter α: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf2.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/pareto_pdf2.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.pareto.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/poisson_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/poisson_dist.html index f053b2d1b..0c82de5f7 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/poisson_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/poisson_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Poisson Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="pareto.html" title="Pareto Distribution"> <link rel="next" href="rayleigh.html" title="Rayleigh Distribution"> @@ -64,7 +64,7 @@ It has the Probability Mass Function: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/poisson_ref1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/poisson_ref1.svg"></span> </p> <p> for k events, with an expected number of events λ. @@ -73,7 +73,7 @@ The following graph illustrates how the PDF varies with the parameter λ: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/poisson_pdf_1.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/poisson_pdf_1.svg" align="middle"></span> </p> <div class="caution"><table border="0" summary="Caution"> <tr> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/rayleigh.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/rayleigh.html index d891caa11..4f5b04fc6 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/rayleigh.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/rayleigh.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Rayleigh Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="poisson_dist.html" title="Poisson Distribution"> <link rel="next" href="skew_normal_dist.html" title="Skew Normal Distribution"> @@ -71,13 +71,13 @@ varies with the shape parameter σ: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/rayleigh_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/rayleigh_pdf.svg" align="middle"></span> </p> <p> and the Cumulative Distribution Function (cdf) </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/rayleigh_cdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/rayleigh_cdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.rayleigh.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/skew_normal_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/skew_normal_dist.html index f030c0572..eb4982442 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/skew_normal_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/skew_normal_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Skew Normal Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="rayleigh.html" title="Rayleigh Distribution"> <link rel="next" href="students_t_dist.html" title="Students t Distribution"> @@ -67,13 +67,13 @@ function is </p> <p> -     <span class="inlinemediaobject"><img src="../../../../equations/normal01_pdf.png"></span> +     <span class="inlinemediaobject"><img src="../../../../equations/normal01_pdf.svg"></span> </p> <p> and the cumulative distribution function </p> <p> -     <span class="inlinemediaobject"><img src="../../../../equations/normal01_cdf.png"></span> +     <span class="inlinemediaobject"><img src="../../../../equations/normal01_cdf.svg"></span> </p> <p> then the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">PDF</a> @@ -82,7 +82,7 @@ Leonhard (1976) is </p> <p> -     <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf0.png"></span> +     <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf0.svg"></span> </p> <p> Given <a href="http://en.wikipedia.org/wiki/Location_parameter" target="_top">location</a> @@ -92,13 +92,13 @@ to the form: </p> <p> -     <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf.png"></span> +     <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf.svg"></span> </p> <p> and <a href="http://en.wikipedia.org/wiki/Cumulative_distribution_function" target="_top">CDF</a>: </p> <p> -     <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_cdf.png"></span> +     <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_cdf.svg"></span> </p> <p> where <span class="emphasis"><em>T(h,a)</em></span> is Owen's T function, and <span class="emphasis"><em>Φ(x)</em></span> @@ -109,8 +109,8 @@ following graphs: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/skew_normal_pdf.png" align="middle"></span> - <span class="inlinemediaobject"><img src="../../../../graphs/skew_normal_cdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/skew_normal_pdf.svg" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/skew_normal_cdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h0"></a> @@ -160,7 +160,7 @@ </p> <p> A parameter δ is related to the shape α by δ = α / (1 + α²), and used in the expression - for skewness <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_skewness.png"></span> + for skewness <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_skewness.svg"></span> </p> </td></tr> @@ -302,7 +302,7 @@ </td> <td> <p> - Using: <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf.png"></span> + Using: <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf.svg"></span> </p> </td> </tr> @@ -314,7 +314,7 @@ </td> <td> <p> - Using: <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_cdf.png"></span><br> where <span class="emphasis"><em>T(h,a)</em></span> + Using: <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_cdf.svg"></span><br> where <span class="emphasis"><em>T(h,a)</em></span> is Owen's T function, and <span class="emphasis"><em>Φ(x)</em></span> is the normal distribution. </p> @@ -412,7 +412,7 @@ </td> <td> <p> - <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_mean.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_mean.svg"></span> </p> </td> </tr> @@ -436,7 +436,7 @@ </td> <td> <p> - <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_variance.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_variance.svg"></span> </p> </td> </tr> @@ -448,7 +448,7 @@ </td> <td> <p> - <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_skewness.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_skewness.svg"></span> </p> </td> </tr> @@ -472,7 +472,7 @@ </td> <td> <p> - <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_kurt_ex.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_kurt_ex.svg"></span> </p> </td> </tr> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/students_t_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/students_t_dist.html index 751b33b42..b0567f767 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/students_t_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/students_t_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Students t Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="skew_normal_dist.html" title="Skew Normal Distribution"> <link rel="next" href="triangular_dist.html" title="Triangular Distribution"> @@ -66,7 +66,7 @@ Given N independent measurements, let </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/students_t_dist.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/students_t_dist.svg"></span> </p> <p> where <span class="emphasis"><em>M</em></span> is the population mean,<span class="emphasis"><em>μ</em></span> @@ -78,7 +78,7 @@ the true standard deviation of the sample. It has the PDF: </p> <p> - <span class="inlinemediaobject"><img src="../../../../equations/students_t_ref1.png"></span> + <span class="inlinemediaobject"><img src="../../../../equations/students_t_ref1.svg"></span> </p> <p> The Student's t-distribution takes a single parameter: the number of degrees @@ -89,7 +89,7 @@ with the degrees of freedom ν: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/students_t_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/students_t_pdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.students_t_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/triangular_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/triangular_dist.html index 01da61fdd..6b74caf06 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/triangular_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/triangular_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Triangular Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="students_t_dist.html" title="Students t Distribution"> <link rel="next" href="uniform_dist.html" title="Uniform Distribution"> @@ -113,13 +113,13 @@ density function PDF</a> varies with the various parameters: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/triangular_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/triangular_pdf.svg" align="middle"></span> </p> <p> and cumulative distribution function </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/triangular_cdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/triangular_cdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.triangular_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/uniform_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/uniform_dist.html index aac50b5c4..0b3a4a072 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/uniform_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/uniform_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Uniform Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="triangular_dist.html" title="Triangular Distribution"> <link rel="next" href="weibull_dist.html" title="Weibull Distribution"> @@ -102,13 +102,13 @@ density function PDF</a> varies with the shape parameter: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/uniform_pdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/uniform_pdf.svg" align="middle"></span> </p> <p> Likewise for the CDF: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/uniform_cdf.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/uniform_cdf.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.uniform_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/dists/weibull_dist.html b/libs/math/doc/html/math_toolkit/dist_ref/dists/weibull_dist.html index a26907460..25cefdd39 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/dists/weibull_dist.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/dists/weibull_dist.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Weibull Distribution</title> <link rel="stylesheet" href="../../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="uniform_dist.html" title="Uniform Distribution"> <link rel="next" href="../dist_algorithms.html" title="Distribution Algorithms"> @@ -84,14 +84,14 @@ α: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/weibull_pdf1.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/weibull_pdf1.svg" align="middle"></span> </p> <p> While this graph illustrates how the PDF varies with the scale parameter β: </p> <p> - <span class="inlinemediaobject"><img src="../../../../graphs/weibull_pdf2.png" align="middle"></span> + <span class="inlinemediaobject"><img src="../../../../graphs/weibull_pdf2.svg" align="middle"></span> </p> <h5> <a name="math_toolkit.dist_ref.dists.weibull_dist.h0"></a> diff --git a/libs/math/doc/html/math_toolkit/dist_ref/nmp.html b/libs/math/doc/html/math_toolkit/dist_ref/nmp.html index 669bb8ad0..5ac329426 100644 --- a/libs/math/doc/html/math_toolkit/dist_ref/nmp.html +++ b/libs/math/doc/html/math_toolkit/dist_ref/nmp.html @@ -3,8 +3,8 @@ <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Non-Member Properties</title> <link rel="stylesheet" href="../../math.css" type="text/css"> -<meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> -<link rel="home" href="../../index.html" title="Math Toolkit 2.1.0"> +<meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> +<link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> <link rel="up" href="../dist_ref.html" title="Statistical Distributions Reference"> <link rel="prev" href="../dist_ref.html" title="Statistical Distributions Reference"> <link rel="next" href="dists.html" title="Distributions"> @@ -273,7 +273,7 @@ if the random variable is outside the defined range for the distribution. </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/hazard.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/hazard.svg"></span> </p> <div class="caution"><table border="0" summary="Caution"> <tr> @@ -302,7 +302,7 @@ if the random variable is outside the defined range for the distribution. </p> <p> - <span class="inlinemediaobject"><img src="../../../equations/chf.png"></span> + <span class="inlinemediaobject"><img src="../../../equations/chf.svg"></span> </p> <div class="caution"><table border="0" summary="Caution"> <tr> |